Reduction of Bundles, Connection, Curvature, and Torsion of the Centered Planes Space at Normalization

• Published in: Mathematics
• Main Author: Olga Belova
• Format: Article in Journal/Newspaper
• Language: English
• Published: MDPI AG 2019
• Subjects: differentiable manifold; cartan–laptev method; space of centered planes; normalization; reduction; connection; Mathematics; QA1-939; laptev
• Online Access: https://doi.org/10.3390/math7100901; https://doaj.org/article/56adf3c717cb45c1ab3b80c881255b70

The space <math display="inline"> <semantics> <mo>Π</mo> </semantics> </math> of centered m -planes is considered in projective space <math display="inline"> <semantics> <msub> <mi>P</mi> <mi>n</mi> </msub> </semantics> </math> . A principal bundle is associated with the space <math display="inline"> <semantics> <mo>Π</mo> </semantics> </math> and a group connection is given on the principal bundle. The connection is not uniquely induced at the normalization of the space <math display="inline"> <semantics> <mo>Π</mo> </semantics> </math> . Semi-normalized spaces <math display="inline"> <semantics> <msup> <mo>Π</mo> <mn>1</mn> </msup> </semantics> </math> , <math display="inline"> <semantics> <msup> <mo>Π</mo> <mn>2</mn> </msup> </semantics> </math> and normalized space <math display="inline"> <semantics> <msup> <mo>Π</mo> <mrow> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </msup> </semantics> </math> are investigated. By virtue of the Cartan−Laptev method, the dynamics of changes of corresponding bundles, group connection objects, curvature and torsion of the connections are discovered at a transition from the space <math display="inline"> <semantics> <mo>Π</mo> </semantics> </math> to the normalized space <math display="inline"> <semantics> <msup> <mo>Π</mo> <mrow> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </msup> </semantics> </math> .